Consider the vector equation.

\[ x_{1} \left[\begin{array}{c} 1 \\ 0 \\ 1 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} 6 \\ 1 \\ -1 \\ 2 \end{array}\right] + x_{3} \left[\begin{array}{c} 5 \\ 2 \\ -4 \\ 2 \end{array}\right] = \left[\begin{array}{c} -7 \\ -1 \\ 0 \\ -2 \end{array}\right] \]

  1. Write a system of scalar equations corresponding to this vector equation.
  2. Write an augmented matrix corresponding to this vector equation.

Answer:

  1. \[\begin{matrix} x & + & 6 \, y & + & 5 \, z & = & -7 \\ & & y & + & 2 \, z & = & -1 \\ x & - & y & - & 4 \, z & = & 0 \\ & & 2 \, y & + & 2 \, z & = & -2 \\ \end{matrix}\]

  2. \[ \left[\begin{array}{ccc|c} 1 & 6 & 5 & -7 \\ 0 & 1 & 2 & -1 \\ 1 & -1 & -4 & 0 \\ 0 & 2 & 2 & -2 \end{array}\right] \]